Signmagnitude

Sign-magnitude is a method for representing signed integers in binary by separating a sign from the magnitude. In an n-bit word, the most significant bit serves as the sign: 0 indicates nonnegative values and 1 indicates negative values. The remaining n−1 bits store the magnitude, which ranges from 0 to 2^(n−1)−1. As a result, the representable set is from −(2^(n−1)−1) to +(2^(n−1)−1), plus two distinct representations of zero: +0 (sign 0, magnitude 0) and −0 (sign 1, magnitude 0).

Example (4-bit sign-magnitude): +3 is 0 011, −3 is 1 011; +7 is 0 111, −7 is 1 111; +0 is 0 000 and −0 is 1 000.

Arithmetic with sign-magnitude numbers is not as straightforward as with two’s complement. Adding numbers requires handling the signs and magnitudes separately, often by comparing magnitudes and assigning the sign of the larger magnitude, or by converting to another representation. The dual zero representations complicate equality testing and range checks. Overflow occurs when a magnitude exceeds the maximum representable value, even though a sign may remain correct.

Compared with two’s complement and one’s complement, sign-magnitude representation is less convenient for hardware arithmetic. Two’s complement collapses the sign and magnitude into a single natural range and yields a single zero, which simplifies addition and subtraction. One’s complement also has a dual zero but uses a different negation rule. Sign-magnitude was used in some early machines and certain fixed-point formats, but it is rarely used for modern integer arithmetic. It remains more a historical or specialized encoding than a practical general-purpose format.